Equations of hyperelliptic Shimura curves

Abstract

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefnite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our algorithm that had been conjectured by Kurihara.

References in corpus (1)

• Shimura curves of genus at most two

Cited by in corpus (6)

• Equations of hyperelliptic Shimura curves
• Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves
• An intriguing hyperelliptic Shimura curve quotient of genus 16
• Quaternionic loci in Siegel's modular threefold
• On the gonality, treewidth, and orientable genus of a graph
• Ribet bimodules and the specialization of Heegner points